Journal of Applied Mathematics and Stochastic Analysis 4, Numr 3, Fall 1991, 259-262
ON A COUNTEREXAMPLE OF GAROFALO LIN FOR A UNIQUE
CONTINUATION OF SCHRDINGER EQUATION*
WILL Y. LEE
Departrnent of Mathematical Sciences
Rutgers University-Camden,
Camden, N. J. 08102
ABSTRACT
Garofalo and Lin have given a counterexample for which
unique continuation fails for the Schr0dinger equation
u +
c
iX-i +
U
0,
> 0.
Their counterexample consists of a Bessel function of the third
with the restriction that v cannot be an integer. In this
note we have removed the restriction.
kind
Kv(Ixl)
Key Words: SchrOdinger Equation, Unique Continuation,
Bessel Function of the Third Kind, Inverse Square Potential
AMS Subject Classification" 35A07, 35L05.
Consider the following Schr6dinger equation:
(1)
-Au + Vu = 0.
Garofalo and Lin [ 1] have shown that V = c/l xl 2, the inverse square potential, is
optimal for unique continuation of (1). Indeed they have given a counterexample for
which unique continuation fails once the square inverse potential is replaced by c/I xl 2+e
for any > 0 [1: pp. 265-266]. Their counterexample is given by
u(x) = lxl
-(n-2)/2
2,/-
K(n-2)/ -e
Ixl-e/21
Received: October 1990, Revised: January 1991
Printed in the U.S.A. (C) 1991 The Society of Applied Mathematics, Modeling and Simulation
259
WILL Y. LEE
260
with the restriction that (n-2)/e cannot be an integer for any e > 0
In this note we show that this restriction is unnecessary. We use the same notations as in
[1].
Theorem: For the following Schrrdinger equation,
-zSu +
(2)
c
u = 0 in B 1,
Ixl2+e
where B 1 is a unit ball in Rn, we have a radial solution given by
(
2ff
u(x) = Ixl’(n’2)/2K(n-2)/ -3 Ixl
(3)
where
K(n_2)/e
-/21
is a Bessel function of the third kindfor any real number (n-2)/e and
e>0.
Proof.
AxU
Since
n-1
= Urr + r Ur
+-1 A0u,
cr-e
r2 u"(r) + (n-l) r u’(r)
(4)
a radial solution of (2) must satisfy
r’-,
u(r) = 0, 0 < r < 1.
It is known that Iv(z) and I_v (z), Bessel functions of imaginary argument, satisfy the
differential equation [ 3" p. 77]
d2 u
z2----+
dz2
(5)
du
(z2 + v2)
u = 0.
Notice from (4) and (5) that Iv (z) and I_v (z) (when v is an integer, see equation (9)
for the definition) cannot be a solution of (4) because of the coefficient n-1 of u’ (r) in
(4). Thus we look for a solution of a form zeq._+v ([3zY), where c, 3, and 7 are
constants to be determined. An easy computation reveals that za I+v ([3zY)) satisfies"
(6)
z2
d2
z
t
z
2 (z I_+v(ZY)) + (1- 2)
d
(’y2ZzZY +
(zaI_+v(ZY))
2V2- Ot2) (ZtI+v(zY))
= O.
On a Counterexample
261
of Garofalo-Lin
Comparison of (6) with (4) shows that
c = -(n- 2)/2, [5 =- 2,f’/e, 7 =-e/2, and v = (n-2)/e.
(7)
Define the third kind of Bessel function Kv(z) according to Watson [ 3: p. 78] by
,.v(Z!. -Iv (z),.,
(8)
Kv(z) = 2
(9)
Kv(z),
Kn(z) = lim
V----)n
sin
v rc
v # integer
v = integer.
Then Kv(z) is defined for all real values of v. Conjunction of equations (4)-(9) yields
the solution (3) to Schrtdinger equation (2). This completes the proof.
Remark.
Since Kv(r)~ r-1/2 e-r for all v as
[ 3: p.202], Kv(r) vanishes of infinite
order as r oo. Consequently unique continuation of Schrtdinger equation (2) fails for
any e > 0, which implies that V = c/I xi 2, the inverse square potential, is optimal for
unique continuation of solutions of Schrtdinger equation (1).
r---) oo
References
Garofalo, N. and Lin, F.H., Mo..not0n0city .Propei.es of..Variational.....!ntegral
[21
AI2
W.ei.ghts and Unique Continuation, Indiana Univ. Math. Jour., Vol. 35, No. 2
(1986), 245-268.
Garofalo, N. and Lin, F.H., Uniqu Continuation for Elliptic Qperators: A
Geometricr.Variationa! Approach, Comm. Pure & Appl. Math., Vol. XL (1987)
347-366.
[3]
Watson, G.N., A Treatise on the Theory of Bessel Functions, Cambridge Univ.
Press (1966).